Classification of algebras (associative, commutative, idempotent) Is there a classification of algebras according to the type of operations they have? Ideally I want to know if there is a list which says which algebras (any we can think of) have operations that are associative, commutative and idempotent. 
It can be any kind of algebra we can think of (I doubt the list is big, and I assume there will be algebra isomorphisms relating some of them). 
If such a classification does not exist, is there some way to obtain information in this direction?
 A: "I doubt the list is big."
That, I'm afraid, is quite wrong.
What you are straining for is Universal Algebra, which seeks to study the general features of algebras.
Given a set $X$, and a nonnegative integer $n$, an $n$-ary operation on $X$ is a function $X^n\to X$. You can also have infinitary operations, where you take an infinite ordinal $\alpha$ and consider a function $X^{\alpha}\to X$.  Given an operation $f$ on $X$, its arity is the $n$ such that $f$ is a function from $X^n$ to $X$. So the arity of addition in $\mathbb{R}$ is $2$, for example, since it is a function $\mathbb{R}^2\to\mathbb{R}$, mapping $(a,b)$ to $a+b$. 
A type is an ordered pair $(\Omega, \mathrm{ari}_{\Omega})$, where $\Omega$ is a set, and $\mathrm{ari}_{\Omega}$ is a map from $\Omega$ to the ordinals. The elements of $\Omega$ are the operation symbols, and the ordinal $\mathrm{ari}_{\omega}(s)$ is the arity of $s$ for each $s\in \Omega$.
If $(\Omega,\mathrm{ari}_{\Omega})$ is a type, then an algebra of type $\Omega$ is a pair $(A, (s_A)_{s\in\Omega})$, where $A$ is a set, and for each $s\in \Omega$, $s_A$ is an $\mathrm{ari}_{\Omega}(s)$-ary operation on $A$.
For instance, a type $(\{+\},\mathrm{ari})$, with $\mathrm{ari}(+)=2$ defines algebras that consist of a set with a binary operation.
Once you have a type, you can impose identities. For instance, in an algebra as above, you can impose the condition that $+(+(a,b),c) = +(a,+(b,c))$, which is associativity, etc.
The collection of all algebras of given type $(\Omega,\mathrm{ari}_{\Omega})$ form a category. The collection of all algebras of a given type that satisfy a given set of identities is called a variety of ($\Omega$-)algebras.  The study of varieties of algebras is one of the main concerns of Universal Algebra.
There are too many of them to discuss individually. In fact, if you just consider how many classes of groups there are that satisfy certain sets of identities, the answer is uncountably many (there are uncountably many varieties of groups); in fact, there are uncountably many varieties of solvable groups (though only countably many varieties of abelian groups). 
I recommend George Bergman's An Invitation to General Algebra and Universal Constructions, Springer Verlag Universitext; or Ralph McKenzie, George McNulty, and Walter Taylor's Algebras, Lattices, Varieties volume 1, Wadsworth and Brooks/Cole; or George Gratzer's Universal Algebra, Springer Verlag. 
